Jacobi inversion formulae for a curve in Weierstrass normal form

TL;DR

This paper derives Jacobi inversion formulae for algebraic curves in Weierstrass normal form, generalizing elliptic functions, and applies Jorgenson's results to analyze the Jacobian strata of such curves.

Contribution

It introduces Jacobi inversion formulae for a broad class of curves in Weierstrass form, extending classical elliptic function theory.

Findings

Derived Jacobi inversion formulae for these curves

Connected the formulae to the structure of the Jacobian strata

Extended classical elliptic function results to higher genus curves

Abstract

We consider a pointed curve $(X,P)$ which is given by the Weierstrass normal form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +\cdots + A_{r-1}(x) y + A_{r}(x)$ where $x$ is an affine coordinate on $\mathbb{P}^1$, the point $\infty$ on $X$ is mapped to $x=\infty$, and each $A_j$ is a polynomial in $x$ of degree $\leq js/r$ for a certain coprime positive integers $r$ and $s$ ($r<s$) so that its Weierstrass non-gap sequence at $\infty$ is a numerical semigroup. It is a natural generalization of Weierstrass' equation in the Weierstrass elliptic function theory. We investigate such a curve and show the Jacobi inversion formulae of the strata of its Jacobian using the result of Jorgenson (Israel J. Math (1992) 77 pp 273-284).